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Quantum gates are unitary transformations that manipulate qubit states, analogous to classical logic gates but reversible and operating on amplitudes rather than bits. Single-qubit gates include the Pauli gates (X, Y, Z), the Hadamard gate (H), and phase gates (S, T). The controlled-NOT (CNOT) is the standard two-qubit gate. Any unitary operation on n qubits can be decomposed into single-qubit gates plus CNOT — this is the universality theorem that makes a finite gate set sufficient for arbitrary quantum computation.
From your study of Pauli matrices and linear transformations, you know that quantum states are vectors and transformations are matrices. Quantum gates are the specific matrices used in quantum computation, and the key constraint is that every gate must be unitary: U*U^dagger = I. Unitarity preserves the norm of the state vector (probabilities sum to 1) and ensures reversibility (every gate has an inverse). This is the quantum analog of classical logic gates, but with a crucial difference — classical gates like AND are irreversible (two inputs, one output), while quantum gates must always be invertible.
The most important single-qubit gates form a small toolkit. The Pauli gates X, Y, Z are the quantum analogs of bit-flip, bit-and-phase-flip, and phase-flip respectively. X swaps |0> and |1> (like a classical NOT). Z leaves |0> alone and maps |1> to -|1>, introducing a relative phase. The Hadamard gate H maps |0> to (|0> + |1>)/sqrt(2) and |1> to (|0> - |1>)/sqrt(2), creating equal superpositions. H is the workhorse that creates interference — nearly every quantum algorithm starts by applying H to each qubit to create a uniform superposition over all computational basis states. The phase gates S and T introduce phases of i and ei*pi/4 respectively to the |1> component; the T gate is particularly important because {H, T, CNOT} forms a universal gate set.
The standard two-qubit gate is the controlled-NOT (CNOT). It has a control qubit and a target qubit: if the control is |1>, the target is flipped; if the control is |0>, nothing happens. In matrix form, it is a 4x4 permutation matrix. The CNOT is the primary entanglement-generating gate — applying H to the first qubit followed by CNOT produces the Bell state (|00> + |11>)/sqrt(2) from |00>. Other important multi-qubit gates include the controlled-Z (CZ), the Toffoli gate (controlled-controlled-NOT), and the SWAP gate, but all can be built from CNOT and single-qubit gates.
The universality theorem states that any unitary operation on n qubits can be approximated to arbitrary precision using only single-qubit gates and CNOT. More precisely, the set {H, T, CNOT} is universal: any n-qubit unitary can be decomposed into a finite sequence of these three gates. This is the quantum analog of the classical fact that {NAND} is a universal gate — but with the added subtlety that quantum universality requires approximation (the Solovay-Kitaev theorem quantifies how efficiently). This means hardware designers need only implement a small set of physical gates, and compilers handle the decomposition of arbitrary operations into this basis.
Understanding quantum gates requires distinguishing global phase from relative phase. Multiplying an entire state by ei*theta changes nothing observable — |psi> and ei*theta|psi> are the same physical state. But a phase applied to one component relative to another is physically meaningful: alpha|0> + beta|1> and alpha|0> - beta|1> are genuinely different states that produce different measurement outcomes after further gates. Gates like Z and S manipulate relative phases, which then get converted to probability differences by subsequent Hadamard gates. This interplay between phase manipulation and interference is the core mechanism of quantum algorithms.